describe the end behavior of the quadratic function graphed below. pick all that apply.\ndraw\nthe function…

Explanation:

Step1: Analyze the parabola's direction

The quadratic function's graph is a parabola opening downward (since it has a maximum point at the vertex). The general form of a quadratic is ( y = ax^2 + bx + c ), and if ( a < 0 ), it opens downward.

Step2: Analyze as ( x \to +\infty )

For a downward - opening parabola, as ( x ) approaches positive infinity (( x\to+\infty )), the ( y ) - values (function values) decrease. So "The function values decrease as ( x ) approaches positive infinity" is correct.

Step3: Analyze as ( x \to -\infty )

For a downward - opening parabola, as ( x ) approaches negative infinity (( x\to-\infty )), the ( y ) - values (function values) decrease? Wait, no. Wait, the vertex is at ( x = 0 ). For a downward - opening parabola ( y=ax^{2}+bx + c,a < 0 ), the left - hand side (as ( x\to-\infty )): let's take the derivative (or just analyze the shape). The parabola opens downward, so as ( x) moves from ( -\infty ) towards the vertex (( x = 0 )), the function values increase, and as ( x) moves from the vertex towards ( +\infty ), the function values decrease. Wait, I made a mistake earlier. Let's re - analyze. The vertex is at ( (0,0) ) (wait, no, looking at the graph, the vertex is at ( (0,0) )? Wait, the graph is a parabola opening downward with vertex at ( (0,0) )? Wait, no, the graph is below the x - axis? Wait, the graph is a parabola opening downward, with vertex at ( (0,0) )? Wait, no, the graph is drawn with the vertex at the origin (0,0) and opening downward. So the equation is ( y=-ax^{2},a>0 ). So as ( x\to+\infty ), ( y=-ax^{2}\to-\infty ) (decreasing), and as ( x\to-\infty ), ( y = - ax^{2}\to-\infty )? Wait, no, that can't be. Wait, no, the graph in the picture: let's look at the grid. The vertex is at (0,0), and the parabola opens downward. So when ( x ) is positive and getting larger (approaching ( +\infty )), the ( y ) - values (function values) decrease (because it's opening downward). When ( x ) is negative and getting more negative (approaching ( -\infty )), the ( y ) - values (function values) decrease? Wait, no, that's not right. Wait, for ( y=-x^{2} ), when ( x = 1 ), ( y=-1 ); ( x = 2 ), ( y=-4 ) (decreasing as ( x\to+\infty )). When ( x=-1 ), ( y = - 1 ); ( x=-2 ), ( y=-4 ) (also decreasing as ( x\to-\infty ))? Wait, no, that's because the parabola is symmetric about the y - axis. Wait, but in the graph, the parabola is opening downward, so both as ( x\to+\infty ) and ( x\to-\infty ), the function values decrease? But that contradicts my initial thought. Wait, no, let's check the options. The options are:

1. The function values increase as ( x ) approaches positive infinity.

2. The function values decrease as ( x ) approaches positive infinity.

3. The function values increase as ( x ) approaches negative infinity.

4. The function values decrease as ( x ) approaches negative infinity.

For a downward - opening parabola (( a<0 )):

• As ( x\to+\infty ), ( y = ax^{2}+bx + c\to-\infty ) (since ( a<0 ) and ( x^{2}\to+\infty )), so the function values decrease. So option 2 is correct.

• As ( x\to-\infty ), ( x^{2}\to+\infty ), and since ( a < 0 ), ( y=ax^{2}+bx + c\to-\infty ). But wait, when moving from ( -\infty ) towards 0 (the vertex), the function values increase (because at ( x = -\infty ), ( y) is a large negative number, and at ( x = 0 ), ( y = 0 )). Wait, I think I confused the direction. Let's take a point: when ( x=-3 ), ( y=-9 ) (if ( y=-x^{2} )), when ( x=-2 ), ( y = - 4 ) (increase), when ( x=-1 ), ( y=-1 ) (increase), when ( x = 0 ), ( y = 0 ) (increase), when ( x = 1 ), ( y=-1 ) (decrease), when ( x = 2 ), ( y=-4 ) (decrease), when ( x = 3 ), ( y=-9 ) (decrease). So as ( x) approaches negative infinity (moving from right to left, going to more negative ( x )), we are moving from ( x = 0 ) to ( x=- \infty ), so the function values decrease? Wait, no, when ( x) goes from ( 0 ) to ( - \infty ), ( x) is becoming more negative, and ( y) goes from ( 0 ) to ( -\infty ), so the function values decrease as ( x) approaches negative infinity? But when ( x) goes from ( - \infty ) to ( 0 ), the function values increase. So the end - behavior (as ( x\to-\infty )): we look at what happens when ( x) gets very large in the negative direction. So as ( x\to-\infty ), ( y=-x^{2}\to-\infty ), so the function values decrease as ( x) approaches negative infinity? Wait, no, that's not correct. Wait, the end - behavior is about what happens as ( x) approaches ( +\infty ) or ( -\infty ), not as ( x) approaches the vertex. So for ( y=-x^{2} ):

• As ( x\to+\infty ), ( y\to-\infty ) (function values decrease).

• As ( x\to-\infty ), ( y\to-\infty ) (function values decrease)? But that can't be, because the parabola is symmetric. Wait, no, ( (-x)^{2}=x^{2} ), so ( y(-x)=y(x) ). So as ( x\to+\infty ) and ( x\to-\infty ), ( y) behaves the same. So both as ( x\to+\infty ) and ( x\to-\infty ), the function values decrease. But let's check the options. The options are:

Option 2: The function values decrease as ( x ) approaches positive infinity. (Correct)

Option 4: The function values decrease as ( x ) approaches negative infinity. (Correct)

Wait, but let's look at the graph again. The graph is a parabola opening downward with vertex at (0,0). So when ( x ) is positive and getting larger (towards ( +\infty )), the graph goes downward (function values decrease). When ( x ) is negative and getting larger in magnitude (towards ( -\infty )), the graph also goes downward (function values decrease). So the correct options are the second and fourth.

Answer:

The function values decrease as ( x ) approaches positive infinity, The function values decrease as ( x ) approaches negative infinity (i.e., the options "The function values decrease as ( x ) approaches positive infinity" and "The function values decrease as ( x ) approaches negative infinity" are the correct ones).